\beginsection{1.5}

Prove $1+1/2+1/4+\cdots+1/2^n=2-1/2^n$ for all natural numbers $n$.

\medskip
For $n=1$ we have $1+1/2=3/2=2-1/2$ so the formula is true for $n=1$.
Now we want to consider the expression
$$1+1/2+1/4+\cdots+1/2^n+1/2^{n+1}$$
All of the terms except the last can be replaced with $2-1/2^n$ therefore
$$1+1/2+1/4+\cdots+1/2^n+1/2^{n+1}=2-1/2^n+1/2^{n+1}=2-(1/2^n)(1-1/2)
=2-1/2^{n+1}$$
Hence the formula is true for $n+1$ whenever it is true for $n$.
Therefore by induction the formula is true for all $n\ge1$.

